Matrix- and tensor-oriented numerical schemes for the evolutionary space-fractional complex Ginzburg--Landau equation

TL;DR

This work tackles the numerical solution of the multidimensional evolutionary space-fractional complex Ginzburg–Landau equation by exploiting matrix- and tensor-oriented techniques that leverage Kronecker structure in the spatial discretization. The authors diagonalize the discretized operators to efficiently compute actions of matrix functions needed by linearly implicit and exponential-type time integrators, avoiding explicit formation of large dense matrices. They develop both 2D and general d-dimensional formulations, including a tensor Tucker-operator framework, and demonstrate substantial speedups (often many orders of magnitude) over vectorized approaches on both CPUs and GPUs across 2D and 3D test problems. The practical impact lies in enabling fast, scalable simulations of complex fractional PDEs on modern hardware, with clear pathways for extending to additional space-fractional models and non-diagonalizable discretizations.

Abstract

In this manuscript, we propose matrix- and tensor-oriented methods for the numerical solution of the multidimensional evolutionary space-fractional complex Ginzburg--Landau equation. After a suitable spatial semidiscretization, the resulting system of ordinary differential equations is time integrated with stiff-resistant schemes. The needed actions of special matrix functions (e.g., inverse, exponential, and the so-called $\varphi$-functions) are efficiently computed in a direct way by exploiting the underlying tensor structure of the task and taking advantage of high performance BLAS and parallelizable pointwise operations. Several numerical experiments in 2D and 3D, where we apply the proposed technique in the context of linearly-implicit and exponential-type schemes, show the reliability and superiority of the approach against the state-of-the-art, allowing to obtain speedups which range from one to two orders of magnitude. Finally, we demonstrate that in our context a single GPU can be effectively exploited to boost the computations both on consumer- and professional-level hardware.

Figures (9)

• Figure 1: Results of Example \ref{['ex:source']} in 2D using lbdf2-m and lbdf2-v, see Section \ref{['sec:ex1_2D']}. Left plot: increasing number of DOF per direction $n=[200,300,400,500,600]$, fixed number of time steps 25. Right plot: increasing number of time steps $[15,20,25,30,35]$, fixed number of DOF per direction $n=400$.
• Figure 2: Results of Example \ref{['ex:nosource']} in 2D using strang-m and strang-v, see Section \ref{['sec:ex2_2D']}. Left plot: increasing number of DOF per direction $n=[400,600,800,1000,1200]$, fixed number of time steps 15. Right plot: increasing number of time steps $[5,10,15,20,25]$, fixed number of DOF per direction $n=800$.
• Figure 3: Level curves of $\lvert u\rvert$ at different times for Example \ref{['ex:nosource']} in 2D with time integrator strang-m, see Section \ref{['sec:ex2_2D']}.
• Figure 4: Results of Example \ref{['ex:source']} in 2D using krogstad-m and krogstad-v, see Section \ref{['sec:ex3_2D']}. Left plot: increasing number of DOF per direction $n=[200,300,400,500,600]$, fixed number of time steps 25. Right plot: increasing number of time steps $[15,20,25,30,35]$, fixed number of DOF per direction $n=400$.
• Figure 5: Results of Example \ref{['ex:source']} in 3D using lbdf2-t and increasing number of time steps $[15,20,25,30,35]$ (fixed number of DOF per direction $n=200$), see Section \ref{['sec:ex1_3D']}. The software architectures have been used with single-precision arithmetic and the hardware is a standard laptop.

Theorems & Definitions (2)

• Example a
• Example b